About Natural/Forced & Transient/Steady-state Responses

(Note) Discussion of transient response and steady-state error is moot if the system does not have stability.

In order to explain stability,

[Total response] = [Natural response] + [Forced response]
= [Homogeneous solution] + [Particular solution]
~= [Transient response] + [Steady-state response]

where ~= "approximately equals to"

As we see above "Natural response" = "Homogeneous solution" ~= "Transient response" and "Forced response" = "Particular solution" ~= "Steady-state response".
(Note) Above relationships are only value for linear systems!!

Are you still confused? Think about a step response of the first order system, if the 1st order system is RC low pass filter and input is 1V step, the output will initially undergo exponential response, but eventually it will converge to 1V in time domain. The output eventually converges 1V because its input (force signal) was 1V. Thus, the steady-state response is the response that eventually converges to a particular value by responding to a forced input!

Let's find it out why Nautal response is not exactly identical to transient response...etc.
The transient response is the sum of the natural and forced response while natural response is large. The steady-state response is also the sum of the natural and forced response, but while the natural response is small. Thus, the transient and steady-state responses are what you actually see on the plot!
The natural and forced responses are the underlying mathematical components of those responses. Since transient and steady-state responses include both natural and forced responses in the responses, they should be defined by associating with "acceptable error"!

HOWEVER, from my experience, people are often using those terms interchangeably. For examle, "natural response" = "transient response" and "forced response" = "steady-state"!

Is it still confusing???

Let's see an example!

Given: C(s) = (2/5)/s + (3/5)/(s+5)
If I take inverse Laplace transform, c(t) = 2/5 + (3/5)exp(-5t).
To be exact, (2/5) in c(t) is called "forced response" and (3/5)exp(-5t) in c(t) is called "natural response" of the system. If you plot c(t) with Matlab, you will see the response at the beginning is the sum of two terms in c(t), but soon the second term (3/5...term) dies out. Thus, transient response is typically associated with "acceptable error". If the output is not within the acceptable error, the response until it is within the error is called "transient". Otherwise, "steady-state response". For example, if I define "acceptable error" as 1mV, then the response before 1mV error is called "transient", otherwise (so within 1mV error) the response is called "steady-state response"!